Stability of the NLS equation with viscosity effect. (English) Zbl 1216.35141

Summary: A nonlinear Schrödinger (NLS) equation with an effect of viscosity is derived from a Korteweg-de Vries (KdV) equation modified with viscosity using the method of multiple time scale. It is well known that the plane-wave solution of the NLS equation exhibits modulational instability phenomenon. In this paper, the modulational instability of the plane-wave solution of the NLS equation modified with viscosity is investigated. The corresponding modulational dispersion relation is expressed as a quadratic equation with complex-valued coefficients. By restricting the modulational wavenumber into the case of narrow-banded spectra, it is observed that a type of dissipation, in this case the effect of viscosity, stabilizes the modulational instability, as confirmed by earlier findings.


35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
Full Text: DOI arXiv EuDML


[1] A. Scott, Ed., Encylopedia of Nonlinear Science, Routledge Taylor & Francis, New York, NY, USA, 2005. · Zbl 1194.42046
[2] T. B. Benjamin and J. E. Feir, “The disintegration of wave trains in deep water,” Journal of Fluid Mechanics, no. 27, pp. 417-430, 1967. · Zbl 0144.47101
[3] G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, Calif, USA, 2nd edition, 1995.
[4] W. Chester, “Resonant oscillations of water waves,” Proceedings of the Royal Society A, vol. 306, pp. 5-22, 1968.
[5] E. Ott and R. N. Sudan, “Damping of solitary waves. I. Theory,” Physics of Fluids, vol. 13, no. 6, pp. 1432-1434, 1970.
[6] C. C. Mei and L. F. Liu, “The damping of surface gravity waves in a bounded liquid,” Journal of Fluid Mechanics, vol. 59, pp. 239-256, 1973. · Zbl 0262.76022
[7] J. W. Miles, “Korteweg-deVries equation modified by viscosity,” Physics of Fluids, vol. 19, no. 7, p. 1063, 1976. · Zbl 0329.76038
[8] D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Philosophical Magazine, vol. 39, no. 5, pp. 422-443, 1895. · JFM 26.0881.02
[9] T. Kakutani and K. Matsuuchi, “Effect of viscosity on long gravity waves,” Journal of the Physical Society of Japan, vol. 39, no. 1, pp. 237-246, 1975. · Zbl 1334.76018
[10] K. Matsuuchi, “Numerical investigations on long gravity waves under the influence of viscosity,” Journal of the Physical Society of Japan, vol. 41, no. 2, pp. 681-687, 1976.
[11] N. J. Zabusky and C. J. Galvin, “Shallow-water waves, the Korteweg-de Vries equation and solitons,” Journal of Fluid Mechanics, vol. 47, pp. 811-824, 1971.
[12] B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train,” Journal of Fluid Mechanics, vol. 83, no. 1, pp. 49-74, 1977.
[13] B. M. Lake and H. C. Yuen, “A note on some water-wave experiments and the comparison of data with theory,” Journal of Fluid Mechanics, vol. 83, no. 1, pp. 75-81, 1977.
[14] C. G. Koop and G. Butler, “An investigation of internal solitary waves in a two-fluid system,” Journal of Fluid Mechanics, vol. 112, pp. 225-251, 1981. · Zbl 0479.76036
[15] K. P. Das, “A Korteweg-de Vries equation modified by viscosity for waves in a channel of uniform but arbitrary cross section,” Physics of Fluids, vol. 28, no. 3, pp. 770-775, 1985. · Zbl 0604.76027
[16] K. P. Das and J. Chakrabarti, “A Korteweg-de Vries equation modified by viscosity for waves in a two-layer fluid in a channel of arbitrary cross section,” Physics of Fluids, vol. 29, no. 3, pp. 661-666, 1986. · Zbl 0597.76101
[17] H. Demiray, “Modulation of nonlinear waves in a viscous fluid contained in an elastice tube,” International Journal of Non-Linear Mechanics, vol. 36, pp. 649-661, 2001. · Zbl 1345.74035
[18] K. G. Tay, Y. Y. Choy, C. T. Ong, and H. Demiray, “Dissipative nonlinear Schrödinger equation with variable coefficient in a stenosed elastic tube filled with a viscous fluid,” International Journal of Engineering, Science and Technology, vol. 2, no. 4, pp. 708-723, 2010.
[19] S. W. Joo, A. F. Messiter, and W. W. Schultz, “Evolution of weakly nonlinear water waves in the presence of viscosity and surfactant,” Journal of Fluid Mechanics, vol. 229, pp. 135-158, 1991. · Zbl 0850.76156
[20] S. P. Cockburn, H. E. Nistazakis, T. P. Horikis, P. G. Kevrekidis, N. P. Proukakis, and D. J. Frantzeskakis, “Matter-wave dark solitons: stochastic versus analytical results,” Physical Review Letters, vol. 104, no. 17, Article ID 174101, 2010.
[21] E. R. Tracy, H. H. Chen, and Y. C. Lee, “Study of quasiperiodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability,” Physical Review Letters, vol. 53, no. 3, pp. 218-221, 1984.
[22] N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of a periodic sequence of picosecond pulses in an optical fiber. Exact solutions,” Soviet Pysics JETP, vol. 89, pp. 1542-1551, 1985.
[23] N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Teoreticheskaya i Matematicheskaya Fizika, vol. 69, no. 2, pp. 189-194, 1986. · Zbl 0625.35015
[24] P. K. Shukla and J. J. Rasmussen, “Modulational instability of short pulses in long optical fibers,” Optics Letters, vol. 11, no. 3, pp. 171-173, 1986.
[25] M. J. Potasek, “Modulation instability in an extended nonlinear Schrödinger equation,” Optics Letters, vol. 12, no. 11, pp. 921-923, 1987.
[26] M. Tanaka, “Maximum amplitude of modulated wavetrain,” Wave Motion, vol. 12, no. 6, pp. 559-568, 1990.
[27] M.-Y. Su and A. W. Green, “Wave breaking and nonlinear instability coupling,” in The Ocean Surface-Wave Breaking, Turbulent Mixing and Radio Probing, Y. Toba and H. Mitsuyasu, Eds., pp. 31-38, Reidel, Boston, Mass, USA, 1985.
[28] W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Physical Review E, vol. 64, no. 1, pp. 016612/1-016612/8, 2001.
[29] V. V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Physical Review A, vol. 65, no. 2, pp. 021602/1-021602/4, 2002.
[30] Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and B. A. Malomed, “On the modulational instability of the nonlinear Schrödinger equation with dissipation,” Physica Scripta, vol. T113, pp. 74-77, 2004.
[31] H. Segur, D. Henderson, J. Carter et al., “Stabilizing the Benjamin-Feir instability,” Journal of Fluid Mechanics, vol. 539, pp. 229-271, 2005. · Zbl 1120.76022
[32] J. T. Bridges and F. Dias, “Enhancement of the Benjamin-Feir instability with dissipation,” Physics of Fluids, vol. 19, no. 10, Article ID 104104, 2007. · Zbl 1182.76085
[33] J. C. Bronski and Z. Rapti, “Modulational instability for nonlinear Schrödinger equations with a periodic potential,” Dynamics of PDE, vol. 2, no. 4, pp. 335-355, 2005. · Zbl 1105.35113
[34] W. P. Hong, “Modulational instability of the higher-order nonlinear Schrödinger equation with fourth-order dispersion and quintic nonlinear terms,” Zeitschrift fur Naturforschung Section A, vol. 61, no. 5-6, pp. 225-234, 2006.
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